TABLE OF CONTENTS Introduction Chapter I. The relation of the structure of an r.e. set to its degree. 1. Post's program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal sets and high degrees. 4. Low degrees, atomless sets, and invariant degree classes. 5. Incompleteness and completeness for noninvariant properties. Chapter II. The structure, automorphisms, and elementary theory of the r.e. sets. 6. Basic facts and splitting theorems. 7. Hh-simple sets. 8. Major subsets and r-maximal sets. 9. Automorphisms of &. 10. The elementary theory of S. Chapter III. The structure of the r.e. degrees. 11. Basic facts. 12. The finite injury priority method. 13. The infinite injury priority method. 14. The minimal pair method and lattice embeddings in R. 15. Cupping and splitting r.e. degrees. 16. Automorphisms and decidability of R.

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Recursively enumerable sets and degrees

Preface- Acknowledgments- Introduction- I Background- Preliminaries- Computability Theory- Kolmogorov Complexity of Finite Strings- Relating Plain and Prefix-Free Complexity- Effective Reals- II Randomness of Sets- Martin-Lof Randomness- Other Notions of Effective Randomness- Algorithmic Randomness and Turing Reducibility- III Relative Randomness- Measures of Relative Randomness- The Quantity of K- and Other Degrees- Randomness-Theoretic Weakness- Lowness for Other Randomness Notions- Effective Hausdorff Dimension- IV Further Topics- Omega as an Operator- Complexity of CE Sets- References- Index

Algorithmic Randomness and Complexity

The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The complexity and the randomness aspect of a set of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion of randomness of a set. Recent research shows that, conversely, concepts and methods originating from randomness enrich computability theory. Covering the basics as well as recent research results, this book provides a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory.

Computability and randomness

This work deals with set-theoretic independence results (independence from the usual set-theoretic ZFC axioms), in particular for problems on the continuum. Consequently, the theory of iterated forcing is developed. The author gives a complete presentation of the theory of proper forcing and its relatives, starting from the beginning and avoiding the metamathematical considerations. In addition to particular consistency results, the author shows methods which can be used for such independence results. Many of these are presented in an "axiomatic" framework (a la Martin's axiom) for this reason. The main aim of this book is to enable a researcher interested in an independence result of the appropriate kind, to have much of the work done for him, thus allowing him to quote general results.

Proper and Improper Forcing

Publisher Summary This chapter discusses integers and topology. Role in topology of certain cardinals is associated with ω. This chapter discusses the problems on first countability, convergence, and separable metrizable spaces. A typical use of these set theoretic cardinals associated with ω involves topologically defined cardinals. Another use of these set theoretic cardinals associated with ω is that certain topological results hold if one of these cardinals equals ω1. The chapter also discusses the set theory, which states an ordinal is the set of smaller ordinals, and a cardinal is an initial ordinal. ω is ω0, and c is 2ω. The chapter also describes sequential and countable compactness. A countable set A of a space X is said to cluster at x ∈ X if each neighborhood of x contains infinitely many points of A, and it is said to converge to x ∈ X if each neighborhood of x contains all but finitely many points of A. A space is called countably compact if each countably infinite set clusters at some point, and it is called sequentially compact if each countably infinite set has an infinite subset that converges somewhere. Moreover, a space X is called subsequential if for every countably infinite A ⊆ X and for every cluster point x of A, there is an infinite subset of A that converges to x.

The Integers and Topology

Internal cohen extensions

Games of infinite length and perfect information have been studied for many years. There are numerous determinacy results for these games, and there is a wide body of work on consequences of their determinacy. Except for games with very special payoff functions, games of infinite length and imperfect information have been little studied. In 1969, David Blackwell [1] introduced a class of such games and proved a determinacy theorem for a subclass. During the intervening time, there has not been much progress in proving the determinacy of Blackwell's games. Orkin [17] extended Blackwell's result to a slightly wider class. Blackwell [2] found a new proof of his own result. Maitra and Sudderth [9, 10] improved Blackwell's result in a different direction from that of Orkin and also generalized to the case of stochastic games. Recently Vervoort [18] has obtained a substantial improvement. Nevertheless, almost all the basic questions have remained open. In this paper we associate with each Blackwell game a family of perfect information games, and we show that the (mixed strategy) determinacy of the former follows from the (pure strategy) determinacy of the latter. The complexity of the payoff function for the Blackwell game is approximately the same as the complexity of the payoff sets for the perfect information games. In particular, this means that the determinacy of Blackwell games with Borel measurable payoff functions follows from the known determinacy of perfect information games with Borel payoff sets.

The determinacy of Blackwell games

https://www.ams.org/jams/1989-02-01/S0894-0347-1989-0955605-X/S0894-0347-1989-0955605-X.pdf

Donald A. Martin

Papers

Internal cohen extensions

The determinacy of Blackwell games

A proof of projective determinacy

Classes of Recursively Enumerable Sets and Degrees of Unsolvability

Mathematical Problems. Lecture Delivered Before the International Congress of Mathematicians at Paris in 1900